Inflation doesn't end everywhere at once, but rather in select, disconnected locations at any given time, while the space between those locations continues to inflate. There should be multiple, enormous regions of space where inflation ends and a hot Big Bang begins, but they can never encounter one another, as they're separated by regions of inflating space. Wherever inflation begins, it is all but guaranteed to continue for an eternity, at least in places.
Where inflation ends for us, we get a hot Big Bang. The part of the Universe we observe is just one part of this region where inflation ended, with more unobservable Universe beyond that. But there are countlessly many regions, all disconnected from one another, with the same exact story.
Unfortunately, as is the problem with String theory, none of these have testable prediction that can push it out of the realm of speculation and into being a true science.